7. Suppose that you open a bank
account and after 3 months you have $200 in the account, and after 7
months you have $400. Let x denote the number of months and y denote
the amount of money you have in the bank. Find a linear equation
which will give you these amounts of money at these times. Use that
equation to predict how much money you would have in the bank after
12 months if the balance continued to grow linearly at this rate.

If x denotes the number of months and y denotes the amount of
money at that time, then having $200 at the end of 3 months
corresponds to the ordered pair

(x1, y1) = (3, 200)

and having $400 after 7 months corresponds to the ordered pair

(x2, y2) = (7, 400)

First we find the slope. We substitute these numbers into the
formula for the slope

which gives us

A slope of $50/mo means that you are putting $50 per month into
the bank. All we need to finish the equation is the y-intercept. For
this we can use the point-slope form of the equation.

y - y1 = m(x - x1)

When we substitute our numbers into the equation we get

y - 200 = 50(x -3)

Remove parentheses

y - 200 = 50x -150

add 200 to both sides.

y = 50x + 50

We see that the
y-intercept is also 50. That means that your bank balance can be
explained by a linear model where you opened your account with $50
and put $50 in each month thereafter.

We can check that this works. After 3 months our equation would
say

y = 50(3) + 50 = 150 + 50 = 200

After 7 months we would get

y = 50(7) + 50 = 350 + 50 = 400

and it checks.

Since there is only one linear equation which will agree with the
two pieces of information, this must be the one. We can use this
equation to predict how much money you would have in the bank after
any number of months. The problem asks us to predict how much money
would have after 12 months. Substitute a 12 for the x in the equation
and we get